Abelian varieties with prescribed embedding degree

Authors:
David Freeman;Peter Stevenhagen;Marco Streng
Affiliations:
University of California, Berkeley;Mathematisch Instituut, Universiteit Leiden;Mathematisch Instituut, Universiteit Leiden
Venue:
ANTS-VIII'08 Proceedings of the 8th international conference on Algorithmic number theory
Year:
2008

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Abstract

We present an algorithm that, on input of a CM-field K, aninteger k ≥ 1, and a prime r ≡ 1 mod k, constructs a q-Weil numberπ ∈ OK corresponding to an ordinary, simple abelian variety A overthe field F of q elements that has an F-rational point of order r andembedding degree k with respect to r. We then discuss how CM-methods over K can be used to explicitly construct A.